Statistics123456789000987654321ghjk.pptx

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1. Statistics

2. Variance Variance is a measurement of the spread between numbers in a data set.

3. Standard Deviation In statistics, the standard deviation is a measure of the amount of variation of a random variable expected about its mean . A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range

4. Raw Scores The definition of a raw score in statistics is an unaltered measurement. Raw scores have not been weighted, manipulated, calculated, transformed, or converted. An entire data set that has been unaltered is a raw data set.

5. Z - Score Z-score is a statistical measure that quantifies the distance between a data point and the mean of a dataset . It's expressed in terms of standard deviations. It indicates how many standard deviations a data point is from the mean of the distribution.

6. Z - Score For a recent final exam in STAT 500, the mean was 68.55 with a standard deviation of 15.45. If you scored an 80%: 𝑍=(80−68.55 )/15.45=0.74 , which means your score of 80 was 0.74 SD above the mean. If you scored a 60%: 𝑍=(60−68.55 )/15.45 =−0.55, which means your score of 60 was 0.55 SD below the mean.

7. Z - Score The scores can be positive or negative. For data that is symmetric (i.e. bell-shaped) or nearly symmetric, a common application of Z-scores for identifying potential outliers is for any Z-scores that are beyond ± 3.

8. Using z-scores to standardise a distribution Every X value in a distribution can be transformed into a corresponding z-score Any normal distribution can be standardized by converting its values into z scores.  Z scores tell you how many standard deviations from the mean each value lies . Converting a normal distribution into a z-distribution allows you to calculate the probability of certain values occurring and to compare different data sets

9. Using z-scores to make comparison we can compare performance [values] in two different distributions, based on their z-scores . Lower z-score means closer to the meanwhile higher means more far away . Positive means to the right of the mean or greater while negative means lower or smaller than the mean

10. Using z-scores to make comparison Jared scored a 92 on a test with a mean of 88 and a standard deviation of 2.7. Jasper scored an 86 on a test with a mean of 82 and a standard deviation of 1.8. Find the Z-scores for Jared's and Jasper's test scores, and use them to determine who did better on their test relative to their class.

11. Using z-scores to make comparison Step 1 : Compute each test score's Z-score using the mean and standard deviation for that test. For Jared's test, the Z-score is: 𝑍=(𝑥−𝜇)/𝜎 = (92−88)/2.7=4/2.7 = 1.48 For Jasper's test, the Z-score is: 𝑍=(𝑥−𝜇)/𝜎 = (86−82)/1.8 = 4/1.8 = 2.22

12. Using z-scores to make comparison Step 2 : Use Z-scores to compare across data sets. Jared's Z-score of 1.48 says that his score of 92 was between 1 and 2 standard deviations above the mean. Jasper's Z-score of 2.22 says that his score of 86 was a bit more than 2 standard deviations above the mean. So, Jasper's score of 86 was relatively higher for his class than Jared's 92 was for his class.

13. Probability Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

14. What are Equally Likely Events? When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die : Getting 3 and 5 on throwing a die Getting an even number and an odd number on a die Getting 1, 2 or 3 on rolling a die are equally likely events, since the probabilities of each event are equal

15. Random sampling Simple random sample Each member of the population has an equal chance of being selected Independent random sample Each member of the population has an equal chance of being selected AND The probability of being selected stays constant from one selection to the next [if more than one individual is selected] i.e. Sampling with replacement

16. Independent Random Sampling Probability of event A =  

17. Probability and Frequency distributions Probability usually involves a population of scores displayed in a frequency distribution graph . What is the probability of obtaining an individual score of less than 3? [i.e. either 1 or 2?] N = 20

18. Probability and the normal distribution In any normal distribution the percentage of values that lie within a specified number of standard deviations from the mean is the same

19. Graphing Probability … 68 – 95 -99.7% Rule of Thumb revisited One standard deviation either side of the mean captures: Approx 68% of our data Mathematically: 68.26% Two standard deviations either side of the mean captures: Approx 95% of our data Mathematically: 95.44% Three standard deviations either side of the mean captures: Approx 99.7% of our data Mathematically: 99.73%

20. 68% – 95% -99.7% Rule of Thumb revisited 68.26% – 95.44% – 99.73% Maths calculation

21. Probability What is the probability that a randomly selected data value in a normal distribution lies more than 1 standard deviation below the mean? p ( z < - 1.00) What is the probability that a randomly selected data value in a normal distribution lies more than 1 standard deviation above the mean? p ( z > 1.00)

22. Calculating probability in a normal distribution When calculating the probability we should calculate the Z-Score Standardise the distribution [z-score calculation ], z = If scores on a test were normally distributed with: mean of = 60, and a standard deviation of = 12, what is the probability [of a randomly selected person who took the test] of a score greater than 84?  

23. Probability using Unit Normal Table The body always corresponds to the larger part of the distribution can be located on the left or the right of the distributions The tail always corresponds to the smaller part of the distribution again, can be located on the left or the right of the distributions

24. Probability using Unit Normal Table

25. Example Information from the department of Motor Vehicles indicates that the average age of licensed drivers is = 45.7 years with a standard deviation of =12.5 years. Assuming that the distribution of drivers’ ages is approximately normal, What proportion of licensed drivers are older than 50 years old? z = = = 0.34 What proportion of licensed drivers are younger than 30 years old? z = = = - 1.26 [so, 30 is 1.26 sds below]